Left-symmetric Superalgebras on Special Linear Lie Superalgebras

TL;DR

This paper investigates the existence and classification of left-symmetric superalgebras on special linear Lie superalgebras, providing complete classifications for certain cases and conjecturing a general criterion based on algebraic structure.

Contribution

It offers a complete classification for ${\mathfrak{sl}}(2|1)$, shows non-existence for ${\mathfrak{sl}}(m|1)$ when $m\geq 3$, and proves existence for ${\mathfrak{sl}}(m+1|m)$ for all $m\geq 1$, advancing understanding of superalgebra structures.

Findings

Complete classification of superalgebras on ${\mathfrak{sl}}(2|1)$

Non-existence of superalgebras on ${\mathfrak{sl}}(m|1)$ for $m\geq 3$

Existence of superalgebras on ${\mathfrak{sl}}(m+1|m)$ for all $m\geq 1$

Abstract

In this paper, we study the existence and classification problems of left-symmetric superalgebras on special linear Lie superalgebras ${\mathfrak{sl}}(m|n)$ with $m\neq n$. The main three results of this paper are: (i) a complete classification of the left-symmetric superalgebras on ${\mathfrak{sl}}(2|1)$, (ii) ${\mathfrak{sl}}(m|1)$ does not admit left-symmetric superalgebras for $m \geq 3$, and (iii) ${\mathfrak{sl}}(m+1|m)$ admits a left-symmetric superalgebra for every $m \geq 1$. To prove these results we combine existing results on the existence and classification of left-symmetric algebras on the Lie algebras ${\mathfrak{gl}}_m$ with a detailed analysis of small representations of the Lie superalgebras ${\mathfrak{sl}}(m|1)$. We also conjecture that ${\mathfrak{sl}}(m|n)$ admits left-symmetric superalgebras if and only if $m = n+1$.